Let f
f
be the function given by f(x)=∣∣x2−3∣∣⋅(x+0.5)(x2−3)(x+0.5)< br /> f
(
x
)
=
|
x
2
−
3
|
⋅
(
x
+
0.5
)
(
x
2
−
3
)
(
x
+
0.5
)
. on which of the following open intervals is f
f
continuous?

The function is continuous on all subsets of real numbers.

Explanation:

The function f(x)=∣x^2−3∣*(x+0.5)*(x^2−3)*(x+0.5) is composed by the multiplication of 4 parts. The first one, ∣x^2−3∣, is continuous for all real numbers because is composed by h(g(x)) = |g(x)| and  g(x) = x^2−3 where both are continuous for all real numbers. The other 3 parts are the polynomials  (x+0.5), (x^2−3) and (x+0.5); all polynomials is continuous for all real numbers. So, f(x), the multiplication of these continuous functions, is in consequence continuous on all subsets of real numbers.